How to use the elastic and inelastic collision calculator
- Choose a collision on one line or in two dimensions.
- Enter the two masses and the velocities in kilograms and meters per second. Object 1 starts on the left in one dimension.
- Set the coefficient of restitution: 1 for a perfectly elastic collision, 0 for a perfectly inelastic one, or anything between.
- Read the velocities after, the momentum before and after, and the kinetic energy lost.
Formula
v1 after = (m1 v1 + m2 v2 + m2 e (v2 - v1)) / (m1 + m2)
v2 after = (m1 v1 + m2 v2 + m1 e (v1 - v2)) / (m1 + m2)
Energy lost = (1/2) (m1 m2 / (m1 + m2)) (1 - e^2) (v1 - v2)^2
Two dimensions: only the parts along the line of centers change
The coefficient of restitution e is the speed of separation divided by the speed of approach. Momentum m1 v1 + m2 v2 is always conserved in the absence of outside forces.
Three collisions
A 2 kg object at 5 m/s hits a 2 kg object at rest, with e = 1. They swap velocities: the first stops and the second moves off at 5 m/s, with no energy lost. A 3 kg object at 4 m/s hits a 1 kg object moving at -2 m/s with e = 0: they stick and move together at 2.5 m/s, losing 13.5 of the 26 J of kinetic energy, or 51.92 percent.
A 1 kg object at 6 m/s hitting a 2 kg object at 1 m/s with e = 0.6 leaves at 0.666667 and 3.666667 m/s. Momentum stays at 8 kg m/s while kinetic energy falls from 19 J to 13.666667 J. In two dimensions, a 1 kg object at 4 m/s hitting an equal mass at rest with the line of centers at 30 degrees and e = 1 leaves at 2 m/s at -60 degrees, while the other leaves at 3.4641 m/s at 30 degrees.
Elastic and inelastic
In an elastic collision kinetic energy is conserved, as with hard steel balls. In an inelastic one some energy turns into heat, sound or deformation, and in a perfectly inelastic collision the two objects move off together at the speed of their center of mass. Real collisions fall between the two, which the coefficient of restitution handles with one number.
Related motion problems are in the kinematics calculator and the projectile motion calculator, and for circular motion see the centripetal force and banked curve calculator.
Two-dimensional collisions
For smooth objects the contact force acts along the line joining their centers. So the part of each velocity along that line is treated as a one-line collision, and the part across it is unchanged. The angle you give is the direction from the center of object 1 to the center of object 2 at the moment of contact, measured counterclockwise from the x axis.
Assumptions and limits
- Objects are rigid, smooth and have no spin. No outside forces act during the brief collision.
- In one dimension object 1 starts on the left of object 2 and must be moving faster to the right for a collision to happen.
- In two dimensions the objects must be approaching along the line of centers, or no collision occurs.
- Masses are above zero. The coefficient of restitution runs from 0 to 1.
Frequently asked questions
What happens in a perfectly inelastic collision?
The two objects stick together and move at the common velocity (m1 v1 + m2 v2) / (m1 + m2). This is the case that loses the largest possible amount of kinetic energy while keeping momentum.
What is the coefficient of restitution?
It is the ratio of the relative speed after the collision to the relative speed before it. It is 1 for a collision with no energy loss and 0 when the objects stay together. Real bouncing balls fall in between.
Is momentum always conserved?
Yes, for an isolated pair of objects, in both elastic and inelastic collisions. Kinetic energy is the quantity that changes, and the page shows both so you can compare.
Why does the calculator say the objects do not collide?
Object 1 has to catch up with object 2, so it must move faster toward it along the line of motion. If it is slower, or moving away, the objects are separating and nothing happens.